Astronomy 2, Spring 2007

Homework Set 2, due April 24 .

1. What is the fundamental cause of the seasons? How would the seasons change if

(a) the earth’s axis of rotation were perpendicular to its orbital plane (the ecliptic plane)?

(b) the earth’s axis were tilted 90⁰ instead of 23 1/2⁰ from the perpendicular to its orbital plane?

2. The length of the sidereal day is the time it takes for the earth to rotate once with respect to the stars. The mean solar day is the average time it takes the earth to rotate with respect to the sun.

(a) Are the lengths of these two kinds of days different? Which is shorter? Explain.

(b) Does your watch tell you what time of night to go out to see a given set of stars, or do you need more information? Explain.

(c) If you could change the length of the day, what would you change it to make the calendar much simpler? Is that the sidereal day or the mean solar day? Explain your answers.

3. One trajectory that can be used to send a spacecraft from earth to Venus is a an elliptical orbit around the sun that has the sun at one focus, its aphelion (greatest distance from the sun) at the earth’s orbit, and its perihelion (closest distance to the sun) at the orbit of Venus. Thus the elliptical orbit just touches the orbits of the earth and Venus. Suppose a spacecraft is launched from the earth on this orbit and reaches its perihelion just as Venus is at that position in its orbit. Answer the following questions.

(a) Draw a diagram showing the orbits of Venus, the earth, and the spacecraft (its complete orbit if it simply returned to earth’s orbit after passing Venus) and the position of the sun. Assume that Venus is 0.70 au (astronomical units) from the sun.

(b) State Kepler’s Second Law. What does it imply is the position in its orbit where the spacecraft is traveling the fastest?

(c) What is the semi-major axis of the orbit of the spacecraft?

(d) Calculate the eccentricity of the spacecraft’s orbit.

(e) State Kepler’s Third Law. Use it to calculate the time in earth years it will take the spacecraft to reach Venus.

4. As the moon goes through its phases, its apparent size does not change much. However, the apparent size of Venus changes dramatically as it goes through its phases.

(a) With the aid of a diagram, show why Venus exhibits the same phases as the moon and why and how its apparent size changes as the phases change.

(b) Assume that Venus is 0.70 au (astronomical units) from the sun. What are the phases of apparent maximum and minimum size? Calculate the ratio of the maximum apparent size to the minimum apparent size of Venus. Note: the apparent size of an object is inversely proportional to its distance.